Evaluating the last missing ingredient for the three-loop quark static potential by differential equations

Abstract

We analytically evaluate the three-loop Feynman integral which was the last missing ingredient for the analytical evaluation of the three-loop quark static potential. To evaluate the integral we introduce an auxiliary parameter $y$, which corresponds to the residual energy in some of the HQET propagators. We construct a differential system for 109 master integrals depending on $y$ and fix boundary conditions from the asymptotic behaviour in the limit $y\to \infty$. The original integral is recovered from the limit $y\to 0$. To solve these linear differential equations we try to find an $\epsilon$-form of the differential system. Though this step appears to be, strictly speaking, not possible, we succeed to find an $\epsilon$-form of all irreducible diagonal blocks, which is sufficient for solving the differential system in terms of an $\epsilon$ expansion. We find a solution up to weight six in terms of multiple polylogarithms and obtain an analytical result for the required three-loop Feynman integral by taking the limit $y\to 0$. As a by-product, we obtain analytical results for some Feynman integrals typical for HQET.

7 pages

References in corpus (20)

• Harmonic Polylogarithms
• Multiloop integrals in dimensional regularization made simple
• Asymptotic expansion of Feynman integrals near threshold
• Differential Equations for Two-Loop Four-Point Functions
• Differential Equations for Feynman Graph Amplitudes
• Algorithm FIRE -- Feynman Integral REduction
• Numerical evaluation of multiple polylogarithms
• FIRE5: a C++ implementation of Feynman Integral REduction
• Two-Loop Master Integrals for ${\gamma }^{\ast }\to 3$ Jets: The planar topologies
• FIESTA 4: optimized Feynman integral calculations with GPU support
• Three-loop static potential
• Static QCD potential at three-loop order
• Two-Loop Master Integrals for ${\gamma }^{\ast }\to 3$ Jets: The non-planar topologies
• FIRE4, LiteRed and accompanying tools to solve integration by parts relations
• Fermionic contributions to the three-loop static potential
• Evaluating single-scale and/or non-planar diagrams by differential equations
• The Two-Loop Master Integrals for $q\bar{q}\to VV$
• Analytic results for two-loop master integrals for Bhabha scattering I
• Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation
• Introducing SummerTime: a package for high-precision computation of sums appearing in DRA method

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